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arXiv:2012.15691

Quantum error-correcting codes from matrix-product codes related to quasi-orthogonal and quasi-unitary matrices

Cao, Meng

Published Dec 31, 2020• 0 citations
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Abstract

Matrix-product codes over finite fields are an important class of long linear codes by combining several commensurate shorter linear codes with a defining matrix over finite fields. The construction of matrix-product codes with certain self-orthogonality over finite fields is an effective way to obtain good $q$-ary quantum codes of large length. This article has two purposes: the first is to summarize some results of this topic obtained by the author of this article and his cooperators in [10-12]; the second is to add some new results on quasi-orthogonal matrices (resp. quasi-unitary matrices), Euclidean dual-containing (resp. Hermitian dual-containing) matrix-product codes and $q$-ary quantum codes derived from these matrix-product codes.

DOI
10.48550/arxiv.2012.15691

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Quantum error-correcting codes from matrix-product codes related to quasi-orthogonal matrices and quasi-unitary matrices.

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Matrix-product codes over finite fields are an important class of long linear codes by combining several commensurate shorter linear codes with a defining matrix over finite fields. The construction of matrix-product codes with certain self-orthogonality over finite fields is an effective way to obtain good qqq-ary quantum codes of large length. This article has two purposes: the first is to summarize some results of this topic obtained by the author of this article and his cooperators in [10-12]; the second is to add some new results on quasi-orthogonal matrices (resp. quasi-unitary matrices), Euclidean dual-containing (resp. Hermitian dual-containing) matrix-product codes and qqq-ary quantum codes derived from these matrix-product codes.
Matrix-product codes over finite fields are an important class of long linear codes by combining several commensurate shorter linear codes with a defining matrix over finite fields. The construction of matrix-product codes with certain self-orthogonality over finite fields is an effective way to obtain good qqq-ary quantum codes of large length. Specifically, it follows from CSS construction (resp. Hermitian construction) that a matrix-product code over Fq\mathbb{F}_{q}Fq​ (resp. Fq2\mathbb{F}_{q^{2}}Fq2​) which is Euclidean dual-containing (resp. Hermitian dual-containing) can produce a qqq-ary quantum code. In order to obtain such matrix-product codes, a common way is to construct quasi-orthogonal matrices (resp. quasi-unitary matrices) as the defining matrices of matrix-product codes over Fq\mathbb{F}_{q}Fq​ (resp. Fq2\mathbb{F}_{q^{2}}Fq2​). The usage of NSC quasi-orthogonal matrices or NSC quasi-unitary matrices in this process enables the minimum distance lower bound of the corresponding quantum codes to reach its optimum. This article has two purposes: the first is to summarize some results of this topic obtained by the author of this article and his cooperators in \cite{Cao2020Constructioncaowang,Cao2020New,Cao2020Constructionof}; the second is to add some new results on quasi-orthogonal matrices (resp. quasi-unitary matrices), Euclidean dual-containing (resp. Hermitian dual-containing) matrix-product codes and qqq-ary quantum codes derived from these newly constructed matrix-product codes.
Recently, the construction of matrix-product codes with certain self-orthogonality is an effective way to obtain good quantum codes. One of the most frequently used means in this process is to construct quasi-unitary matrices as the defining matrices of matrix-product codes. Further, if the defining matrix is NSC, then the corresponding matrix-product code has maximal minimum distance. Inspired by these idea, we propose a constructive method about NSC quasi-unitary matrices over Fq2 with q being a prime power, and construct k × k NSC quasi-unitary matrices for any kq[x1, . . . , xk], extending or improving the known results. Through such defining matrices, we obtain new classes of quantum codes from matrix-product codes, many of which show better parameters than previously known.
Quasi-cyclic codes have been recently employed in the constructions of quantum error-correcting codes. In this paper, we propose a construction of infinite families of quasi-cyclic codes over $\F_q$ which are self-orthogonal with respect to the Euclidean and Hermitian inner products. In particular, their dimension and a lower bound for their minimum distance are computed using their constituent codes defined over field extensions of Fq\mathbb{F}_qFq​. We also show that the lower bound for the minimum distance satisfies the square-root-like lower bound and also show how dual-containing and self-dual quasi-cyclic codes can arise from our construction. Using the CSS construction, we show the existence of quantum error-correcting codes with good parameters.
The matrix-product (MP) code CA,k:=[C1,C2,…,Ck]⋅A\mathcal{C}_{A,k}:=[\mathcal{C}_{1},\mathcal{C}_{2},\ldots,\mathcal{C}_{k}]\cdot ACA,k​:=[C1​,C2​,…,Ck​]⋅A with a non-singular by column (NSC) matrix AAA plays an important role in constructing good quantum error-correcting codes. In this paper, we study the MP code when the defining matrix AAA satisfies the condition that AA†AA^†AA† is (D,τ)(D,τ)(D,τ)-monomial. We give an explicit formula for calculating the dimension of the Hermitian hull of a MP code. We provide the necessary and sufficient conditions that a MP code is Hermitian dual-containing (HDC), almost Hermitian dual-containing (AHDC), Hermitian self-orthogonal (HSO), almost Hermitian self-orthogonal (AHSO), and Hermitian LCD, respectively. We theoretically determine the number of all possible ways involving the relationships among the constituent codes to yield a MP code with these properties, respectively. We give alternative necessary and sufficient conditions for a MP code to be AHDC and AHSO, respectively, and show several cases where a MP code is not AHDC or AHSO. We provide the construction methods of HDC and AHDC MP codes, including those with optimal minimum distance lower bounds.
The σσσ hulls of matrix-product codes and related entanglement-assisted quantum error-correcting codes
Let SLAut(Fqn)\mathrm{SLAut}(\mathbb{F}_{q}^{n})SLAut(Fqn​) denote the group of all semilinear isometries on Fqn\mathbb{F}_{q}^{n}Fqn​, where q=peq=p^{e}q=pe is a prime power. Matrix-product (MP) codes are a class of long classical codes generated by combining several commensurate classical codes with a defining matrix. We give an explicit formula for calculating the dimension of the σσσ hull of a MP code. As a result, we give necessary and sufficient conditions for the MP codes to be σσσ dual-containing and σσσ self-orthogonal. We prove that dimFq(Hullσ(C))=dimFq(Hullσ(C⊥σ))\mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_σ(\mathcal{C}))=\mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_σ(\mathcal{C}^{\bot_σ}))dimFq​​(Hullσ​(C))=dimFq​​(Hullσ​(C⊥σ​)). We prove that for any integer hhh with max{0,k1−k2}≤h≤dimFq(C1∩C2⊥σ)\mathrm{max}\{0,k_{1}-k_{2}\}\leq h\leq \mathrm{dim}_{\mathbb{F}_{q}}(\mathcal{C}_{1}\cap\mathcal{C}_{2}^{\bot_σ})max{0,k1​−k2​}≤h≤dimFq​​(C1​∩C2⊥σ​​), there exists a linear code C2,h\mathcal{C}_{2,h}C2,h​ monomially equivalent to C2\mathcal{C}_{2}C2​ such that dimFq(C1∩C2,h⊥σ)=h\mathrm{dim}_{\mathbb{F}_{q}}(\mathcal{C}_{1}\cap\mathcal{C}_{2,h}^{\bot_σ})=hdimFq​​(C1​∩C2,h⊥σ​​)=h, where Ci\mathcal{C}_{i}Ci​ is an [n,ki]q[n,k_{i}]_{q}[n,ki​]q​ linear code for i=1,2i=1,2i=1,2. We show that given an [n,k,d]q[n,k,d]_{q}[n,k,d]q​ linear code C\mathcal{C}C, there exists a monomially equivalent [n,k,d]q[n,k,d]_{q}[n,k,d]q​ linear code Ch\mathcal{C}_{h}Ch​, whose σσσ dual code has minimum distance d′d'd′, such that there exist an [[n,k−h,d;n−k−h]]q[[n,k-h,d;n-k-h]]_{q}[[n,k−h,d;n−k−h]]q​ EAQECC and an [[n,n−k−h,d′;k−h]]q[[n,n-k-h,d';k-h]]_{q}[[n,n−k−h,d′;k−h]]q​ EAQECC for every integer hhh with 0≤h≤dimFq(Hullσ(C))0\leq h\leq \mathrm{dim}_{\mathbb{F}_{q}}(\mathrm{Hull}_σ(\mathcal{C}))0≤h≤dimFq​​(Hullσ​(C)). Based on this result, we present a general construction method for deriving EAQECCs with flexible parameters from MP codes related to σσσ hulls.
In 2019, Carletet al. introduced the concept of σ duals of linear codes involving the σ inner product, which generalizes the Euclidean, Hermitian and ℓ-Galois cases. This paper focuses on constructing new and improved classical codes and quantum codes within the framework of the σ inner product. We derive some general properties of linear codes, including matrix-product (MP) codes, with respect to the σ inner product. We develop general methods and design effective routes involving certain optimization problems for constructing σ self-orthogonal (SO) and σ dual-containing (DC) MP codes. Our schemes efficiently generate numerous such codes with new or optimal parameters. We establish the σ construction of quantum stabilizer codes from classical codes. We propose a unified method for constructing two general classes of entanglement-assisted quantum error-correcting codes (EAQECCs) based on the σ hulls of general linear codes. This further yields six types of EAQECCs with flexible parameters based on propagation rules using MP codes under the Euclidean and Hermitian cases. Compared to the best-known ternary EAQECCs, we obtain 17 new ones and 13 of them have improved parameters. Finally, we present two infinite families of q-ary EAQECCs with lengths (q2− 1)(q+ 2) andq2(q+2), respectively. These families include many q-ary QECCs that are not only new according to Grassl’s online database but also surpass those listed in Edel’s online database.